Post on 29-May-2020
Lecture III
Wayne HuTenerife, November 2007
CMB Polarization
Polarized Landscape100
10
10 100 1000
1
0.1
0.01
l (multipole)
∆ (µ
K)
reionization
gravitationalwaves
gravitationallensing
ΘE
EE
BB
EE
Hu & Dodelson (2002)
Recent Data
Ade et al (QUAD, 2007)
Why is the CMB polarized?
Polarization from Thomson Scattering
• Differential cross section depends on polarization and angle
dσdΩ
=3
8π|ε̂′ · ε̂|2σT
dσdΩ
=3
8π|ε̂′ · ε̂|2σT
Polarization from Thomson Scattering
• Isotropic radiation scatters into unpolarized radiation
Polarization from Thomson Scattering
• Quadrupole anisotropies scatter into linear polarization
aligned withcold lobe
Whence Quadrupoles?• Temperature inhomogeneities in a medium• Photons arrive from different regions producing an anisotropy
hot
hot
cold
(Scalar) Temperature InhomogeneityHu & White (1997)
CMB Anisotropy• WMAP map of the CMB temperature anisotropy
Whence Polarization Anisotropy?• Observed photons scatter into the line of sight • Polarization arises from the projection of the quadrupole on the transverse plane
Polarization Multipoles• Mathematically pattern is described by the tensor (spin-2) spherical harmonics [eigenfunctions of Laplacian on trace-free 2 tensor]
• Correspondence with scalar spherical harmonics established via Clebsch-Gordan coefficients (spin x orbital)
• Amplitude of the coefficients in the spherical harmonic expansion are the multipole moments; averaged square is the power
E-tensor harmonic
l=2, m=0
Modulation by Plane Wave
• Amplitude modulated by plane wave → higher multipole moments• Direction detemined by perturbation type → E-modes
Scal
ars
π/2
φ l
0.5
1.0
Polarization Pattern Multipole PowerB/E=0
A Catch-22• Polarization is generated by scattering of anisotropic radiation• Scattering isotropizes radiation• Polarization only arises in optically thin conditions: reionization and end of recombination
• Polarization fraction is at best a small fraction of the 10-5 anisotropy: ~10-6 or µK in amplitude
Reionization
Temperature Inhomogeneity• Temperature inhomogeneity reflects initial density perturbation on large scales• Consider a single Fourier moment:
Locally Transparent• Presently, the matter density is so low that a typical CMB photon will not scatter in a Hubble time (~age of universe)
recombination
observer
transparent
Reversed Expansion• Free electron density in an ionized medium increases as scale factor a-3; when the universe was a tenth of its current size CMB photons have a finite (~10%) chance to scatter
recombination
rescattering
Polarization Anisotropy• Electron sees the temperature anisotropy on its recombination surface and scatters it into a polarization
recombination
polarization
Temperature Correlation• Pattern correlated with the temperature anisotropy that generates it; here an m=0 quadrupole
WMAP 3year
Page et al (WMAP, 2006)
1 10 100 1000Multipole moment (l)
0.01
0.10
1.00
10.00
100.00{l
(l+1)
Cl /
2π
} 1/2
[μK]
Why Care?• Early ionization is puzzling if due to ionizing radiation from normal stars; may indicate more exotic physics is involved
• Reionization screens temperature anisotropy on small scales making the true amplitude of initial fluctuations larger by eτ
• Measuring the growth of fluctuations is one of the best ways of determining the neutrino masses and the dark energy
• Offers an opportunity to study the origin of the low multipole statistical anomalies
• Presents a second, and statistically cleaner, window on gravitational waves from the early universe
Polarization Power Spectrum• Most of the information on ionization history is in the polarization (auto) power spectrum - two models with same optical depth but different ionization fraction
Kaplinghat et al (2002) [figure: Hu & Holder (2003)]
partial ionization
500µ
K'
50µK
'
step
l(l+
1)C
lEE/2
π 10-13
10-14
l10 100
Principal Components• Information on the ionization history is contained in ~5 numbers - essentially coefficients of first few Fourier modes
Hu & Holder (2003) z5 10
(a) Best
(b) Worst
15 20
δxδx
-0.5
0
0.5
-0.5
0
0.5
Representation in Modes• Reproduces the power spectrum and net optical depth (actual τ=0.1375 vs 0.1377); indicates whether multiple physical mechanisms suggested
Hu & Holder (2003)l
truesum modesfiducial
10 100
10-13
10-14
l(l+1
)ClE
E /2π z
x
0
0.4
0.8
10 15 20 25
• Quadrupole in polarization originates from a tight range of scales around the current horizon • Quadrupole in temperature gets contributions from 2 decades in scale
Hu & Okamoto (2003)
Temperature v. Polarization
temperature
polarization
k (Mpc-1)0.01 0.10.0010.0001
3
2
1
0.01
0.006
0.002
(x30)
(x300)wei
ght i
n po
wer
recomb
reionization
ISW
Alignments
Dvorkin, Peiris, Hu (2007)
Quadrupole Octopole
Temperature
E-polarization
Polarization Peaks
Acoustic Oscillations• When T>3000K, medium ionized• Photons tightly coupled to free electrons via Thomson scattering; electrons to protons via Coulomb interactions
• Medium behaves as a perfect fluid • Radiation pressure competes with gravitational attraction causing perturbations to oscillate
Quadrupoles at Recombination's End
• Acoustic inhomongeneities become anisotropies by streaming/diffusion
Quadrupoles at Recombination's End
• Electron "observer" sees a quadrupole anisotropy• Polarization pattern is a projection quadrupole anisotropy
Fluid Imperfections• Perfect fluid: no anisotropic stresses due to scattering
isotropization; baryons and photons move as single fluid
• Fluid imperfections are related to the mean free path of thephotons in the baryons
λC = τ̇−1 where τ̇ = neσT a
is the conformal opacity to Thomson scattering
• Dissipation is related to the diffusion length: random walkapproximation
λD =√
NλC =√
η/λC λC =√
ηλC
the geometric mean between the horizon and mean free path
• λD/η∗ ∼ few %, so expect the peaks >3 to be affected bydissipation
Viscosity & Heat Conduction• Both fluid imperfections are related to the gradient of the velocity
kvγ by opacity τ̇ : slippage of fluids vγ − vb.
• Viscosity is an anisotropic stress or quadrupole moment formed byradiation streaming from hot to cold regions
m=0
v
hot
hot
cold
v
Dimensional Analysis• Viscosity= quadrupole anisotropy that follows the fluid velocity
πγ ≈k
τ̇vγ
• Mean free path related to the damping scale via the random walkkD = (τ̇ /η∗)
1/2 → τ̇ = k2Dη∗• Damping scale at ` ∼ 1000 vs horizon scale at ` ∼ 100 so
kDη∗ ≈ 10
• Polarization amplitude rises to the damping scale to be ∼ 10% ofanisotropy
πγ ≈k
kD
1
10vγ ∆P ≈
`
`D
1
10∆T
• Polarization phase follows fluid velocity
Damping & Polarization
• Quadrupole moments: damp acoustic oscillations from fluid viscosity generates polarization from scattering
• Rise in polarization power coincides with fall in temperature power – l ~ 1000
105 15 20
Ψ
Θ+Ψ
πγ
ks/π
damping
driving
polarization
Acoustic Polarization• Gradient of velocity is along direction of wavevector, so
polarization is pure E-mode
• Velocity is 90◦ out of phase with temperature – turning points ofoscillator are zero points of velocity:
Θ + Ψ ∝ cos(ks); vγ ∝ sin(ks)
• Polarization peaks are at troughs of temperature power
Cross Correlation• Cross correlation of temperature and polarization
(Θ + Ψ)(vγ) ∝ cos(ks) sin(ks) ∝ sin(2ks)
• Oscillation at twice the frequency
• Correlation: radial or tangential around hot spots
• Partial correlation: easier to measure if polarization data is noisy,harder to measure if polarization data is high S/N or if bands donot resolve oscillations
• Good check for systematics and foregrounds
• Comparison of temperature and polarization is proof againstfeatures in initial conditions mimicking acoustic features
Temperature and Polarization Spectra
100
10
10 100 1000
1
0.1
0.01
l (multipole)
∆ (µ
K)
reionization
gravitationalwaves
gravitationallensing
ΘE
EE
BB
EE
Recent Data
Ade et al (QUAD, 2007)
Why Care?• In the standard model, acoustic polarization spectra uniquely predicted by same parameters that control temperature spectra
• Validation of standard model• Improved statistics on cosmological parameters controlling peaks
• Polarization is a complementary and intrinsically more incisive probe of the initial power spectrum and hence inflationary (or alternate) models
• Acoustic polarization is lensed by the large scale structure into B-modes
• Lensing B-modes sensitive to the growth of structure and hence neutrino mass and dark energy
• Contaminate the gravitational wave B-mode signature
Transfer of Initial Power
Hu & Okamoto (2003)
Gravitational Waves
Gravitational Waves• Inflation predicts near scale invariant spectrum of gravitational waves• Amplitude proportional to the square of the Ei=V1/4 energy scale• If inflation is associated with the grand unification Ei~1016 GeV and potentially observable
transverse-tracelessdistortion
Gravitational Wave Pattern• Projection of the quadrupole anisotropy gives polarization pattern• Transverse polarization of gravitational waves breaks azimuthal symmetry
density
perturbation
gravitational
wave
Electric & Magnetic Polarization(a.k.a. gradient & curl)
Kamionkowski, Kosowsky, Stebbins (1997)Zaldarriaga & Seljak (1997)
• Alignment of principal vs polarization axes (curvature matrix vs polarization direction)
E
B
Patterns and Perturbation Types
Kamionkowski, Kosowski, Stebbins (1997); Zaldarriaga & Seljak (1997); Hu & White (1997)
• Amplitude modulated by plane wave → Principal axis• Direction detemined by perturbation type → Polarization axis
Scal
ars
Vec
tors
Tens
ors
π/2
0 π/4 π/2
π/2
φ
θ
10 100l
0.5
1.0
0.5
1.0
0.5
1.0
Polarization Pattern Multipole PowerB/E=0
B/E=6
B/E=8/13
Scaling with Inflationary Energy Scale• RMS B-mode signal scales with inflationary energy scale squared Ei2
1
10 100 1000
0.1
0.01
∆B (µ
K)
l
1
3
0.3
E i (1
016 G
eV)
g-waves
Contamination for Gravitational Waves• Gravitational lensing contamination of B-modes from gravitational waves cleaned to Ei~0.3 x 1016 GeV
Hu & Okamoto (2002) limits by Knox & Song (2002); Cooray, Kedsen, Kamionkowski (2002)
1
10 100 1000
0.1
0.01
∆B (µ
K)
l
g-lensing
1
3
0.3
E i (1
016 G
eV)
g-waves
The B-Bump• Rescattering of gravitational wave anisotropy generates the B-bump• Potentially the most sensitive probe of inflationary energy scale• Potentially enables test of consistency relation (slow roll)
1
10
10 100 1000
l
∆P (µ
K) EE
BBreionization
B-bumprecombination
B-peaklensing
contaminant
Slow Roll Consistency Relation
Mortonson & Hu (2007)
• Consistency relation between tensor-scalar ratio and tensor tilt r = -8nt tested by reionization • Reionization uncertainties controlled by a complete p.c. analysis
instp.c.
Temperature and Polarization Spectra
100
10
10 100 1000
1
0.1
0.01
l (multipole)
∆ (µ
K)
reionization
gravitationalwaves
gravitationallensing
ΘE
EE
BB
EE
Gravitational Lensing
Gravitational Lensing• Gravitational lensing by large scale structure distorts the observed temperature and polarization fields
• Exaggerated example for the temperature
Original Lensed
Polarization Lensing
Polarization Lensing• Since E and B denote the relationship between the polarization amplitude and direction, warping due to lensing creates B-modes
Original Lensed BLensed E
Zaldarriaga & Seljak (1998) [figure: Hu & Okamoto (2001)]
Reconstruction from Polarization• Lensing B-modes correlated to the orignal E-modes in a specific way
• Correlation of E and B allows for a reconstruction of the lens• Reference experiment of 4' beam, 1µK' noise and 100 deg2
Original Mass Map Reconstructed Mass MapHu & Okamoto (2001) [iterative improvement Hirata & Seljak (2003)]
Why Care• Gravitational lensing sensitive to amount and hence growth of
structure
• Examples: massive neutrinos - d lnCBB` /dmν ≈ −1/3eV, darkenergy - d lnCBB` /dw ≈ −1/8
• Mass reconstruction measures the large scale structure on largescales and the mass profile of objects on small scales
• Examples: large scale decontamination of the gravitational wave Bmodes; lensing by SZ clusters combined with optical weak lensingcan make a distance ratio test of the acceleration
Weighing Neutrinos• Massive neutrinos suppress power strongly on small scales
[∆P/P ≈–8Ων/Ωm]: well modeled by [ceff2=wg, cvis2=wg, wg: 1/3→1]
• Degenerate with other effects [tilt n, Ωmh2...] • CMB signal small but breaks degeneracies • 2σ Detection: 0.3eV [Map (pol) + SDSS]
Power Suppression Complementarity
k (h Mpc-1)
mv = 0 eVmv = 1 eV
P(k)
0.01 –0.01 0.0
0.6
0.8
1.0
1.2
1.4
0.01 0.02 0.030.1
0.1
1 SDSS
Ωνh2 = mν/94eV
n
SDSSonly
MAP only
Joint
Hu, Eisenstein, & Tegmark (1998); Eisenstein, Hu & Tegmark (1998)
Lecture III: Summary• Polarization by Thomson scattering of quadrupole anisotropy• Quadrupole anisotropy only sustained in optically thin conditions
of reionization and the end of recombination
• Reionization generates E-modes at low multipoles from andcorrelated to the Sachs-Wolfe anisotropy
• Reionization polarization enables study of ionization history, lowmultipole anomalies, gravitational waves
• Dissipation of acoustic waves during recombination generatesquadrupoles and correlated polarization peaks
• Recombination polarization provides consistency checks, featuresin power spectrum, source of graviational lensing B modes
• Gravitational waves B-mode polarization sensitive to inflationenergy scale and tests slow roll consistency relation
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